scholarly journals Study of Stability of Rotational Motion of Spacecraft with Canonical Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
William Reis Silva ◽  
Maria Cecília F. P. S. Zanardi ◽  
Regina Elaine Santos Cabette ◽  
Jorge Kennety Silva Formiga
Keyword(s):  
Normal Form ◽  
Rotational Motion ◽  
Equilibrium Points ◽  
Gravity Gradient ◽  
Artificial Satellites ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Andoyer Variables ◽  
The Stability ◽  
Orbital Data

This work aims to analyze the stability of the rotational motion of artificial satellites in circular orbit with the influence of gravity gradient torque, using the Andoyer variables. The used method in this paper to analyze stability is the Kovalev-Savchenko theorem. This method requires the reduction of the Hamiltonian in its normal form up to fourth order by means of canonical transformations around equilibrium points. The coefficients of the normal Hamiltonian are indispensable in the study of nonlinear stability of its equilibrium points according to the three established conditions in the theorem. Some physical and orbital data of real satellites were used in the numerical simulations. In comparison with previous work, the results show a greater number of equilibrium points and an optimization in the algorithm to determine the normal form and stability analysis. The results of this paper can directly contribute in maintaining the attitude of artificial satellites.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

scholarly journals Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Dahlia Khaled Bahlool ◽  
Huda Abdul Satar ◽  
Hiba Abdullah Ibrahim
Keyword(s):  
Equilibrium Points ◽  
Global Dynamics ◽  
Persistence Conditions ◽  
Local Bifurcation Analysis ◽  
Uniqueness Of The Solution ◽  
Harvesting Process ◽  
Predator Model ◽  
The Stability ◽  
Predator System ◽  
Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

scholarly journals Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

On the stability of a mathematical model for HIV(AIDS) - cancer dynamics

2021 ◽  
Vol 8 (4) ◽  
pp. 783-796
Author(s):  
H. W. Salih ◽  
◽  
A. Nachaoui ◽  
Keyword(s):  
Mathematical Model ◽  
Highly Active Antiretroviral Therapy ◽  
Lyapunov Method ◽  
Equilibrium Points ◽  
Hiv Treatment ◽  
Biochemical Processes ◽  
Highly Active ◽  
Biological Phenomena ◽  
The Stability ◽  
The Impact

In this work, we study an impulsive mathematical model proposed by Chavez et al. [1] to describe the dynamics of cancer growth and HIV infection, when chemotherapy and HIV treatment are combined. To better understand these complex biological phenomena, we study the stability of equilibrium points. To do this, we construct an appropriate Lyapunov function for the first equilibrium point while the indirect Lyapunov method is used for the second one. None of the equilibrium points obtained allow us to study the stability of the chemotherapeutic dynamics, we then propose a bifurcation of the model and make a study of the bifurcated system which contributes to a better understanding of the underlying biochemical processes which govern this highly active antiretroviral therapy. This shows that this mathematical model is sufficiently realistic to formulate the impact of this treatment.

Bifurcation and stability of resonance of electric vehicle powertrain: Focus on the influence of electromagnetic parameters

2021 ◽  
pp. 095440702110450
Author(s):  
Qingzhen Han ◽  
Shiqin Niu ◽  
Lei He
Keyword(s):  
Electric Vehicle ◽  
Electromechanical Coupling ◽  
Equilibrium Points ◽  
Resonance Curve ◽  
Drive Shaft ◽  
Electromagnetic Parameters ◽  
Stability And Bifurcation ◽  
Fold Bifurcation ◽  
The Stability ◽  
Vehicle Powertrain

The influence of the electromagnetic parameters on the torsional dynamics of the electric vehicle powertrain is studied by considering the electromechanical coupling effect. By adding the electromagnetic torque on the drive side, the powertrain is simplified as nonlinear drive-shaft model. The number, stability, and bifurcation conditions of the equilibrium points of the nonlinear drive-shaft model are deduced. Based on the averaged equations and the amplitude-frequency response equation, the stability and bifurcation conditions, such as fold bifurcation and Hopf bifurcation, of the resonance curve are discussed. The influence of electromagnetic parameters on the torsional dynamics is studied by simulation. It is shown that with the change of the parameters, the number as well as the stability of the equilibrium points may be changed which is affected by fold bifurcation. It is also shown that the resonance curve may lose its stability when fold bifurcation happens. By limiting the parameters in the region without fold bifurcation, the unstable dynamics of the resonance curve can be controlled.

scholarly journals Retrograde resonances in compact multi-planetary systems: a feasible stabilizing mechanism

2007 ◽  
Vol 3 (S249) ◽  
pp. 511-516 ◽  
Author(s):  
Julie Gayon ◽  
Eric Bois
Keyword(s):  
Body Problem ◽  
Extrasolar Planets ◽  
Planetary Systems ◽  
Central Star ◽  
Point Of View ◽  
Outer Planet ◽  
Mean Motion Resonances ◽  
Hot Jupiters ◽  
The Stability ◽  
Orbital Data

AbstractMulti-planet systems detected until now are in most cases characterized by hot-Jupiters close to their central star as well as high eccentricities. As a consequence, from a dynamical point of view, compact multi-planetary systems form a variety of the general N-body problem (with N ≥ 3), whose solutions are not necessarily known. Extrasolar planets are up to now found in prograde (i.e. direct) orbital motions about their host star and often in mean-motion resonances (MMR). In the present paper, we investigate a theoretical alternative suitable for the stability of compact multi-planetary systems. When the outer planet moves on a retrograde orbit in MMR with respect to the inner planet, we find that the so-called retrograde resonances present fine and characteristic structures particularly relevant for dynamical stability. We show that retrograde resonances and their resources open a family of stabilizing mechanisms involving specific behaviors of apsidal precessions. We also point up that for particular orbital data, retrograde MMRs may provide more robust stability compared to the corresponding prograde MMRs.

A discussion on orbital analysis - Determination of the Earth’s gravitational field from satellite orbits: methods and results

1967 ◽  
Vol 262 (1124) ◽  
pp. 119-134 ◽  
Keyword(s):  
Gravitational Field ◽  
Artificial Satellites ◽  
Satellite Orbits ◽  
Estimation Of Parameters ◽  
Earth’S Gravitational Field ◽  
Keplerian Ellipse ◽  
Orbital Data ◽  
Analysis Determination ◽  
Orbital Analysis

The structure of theories used in determining the gravitational field from the perturbations of orbits of artificial satellites is discussed and it is shown how it corresponds to the fact that small departures from a Keplerian ellipse are readily observed. Some current problems are mentioned. Statistical problems in the estimation of parameters of the field from orbital data are considered and recent estimates are summarized

Numerical Investigation of the Influence of Friction Coefficient on Brake Groan

2010 ◽  
Vol 44-47 ◽  
pp. 1923-1927 ◽  
Author(s):  
Xian Jie Meng
Keyword(s):  
Nonlinear Dynamics ◽  
Friction Coefficient ◽  
Degrees Of Freedom ◽  
Equilibrium Points ◽  
Static Friction ◽  
Kinetic Friction ◽  
Dynamics Model ◽  
Excited Vibration ◽  
Nonlinear Dynamics Model ◽  
The Stability

A two degrees of freedom nonlinear dynamics model of self-excited vibration induced by dry-friction of brake disk and pads is built firstly, the stability of vibration system at the equilibrium points is analyzed using the nonlinear dynamics theory. Finally the numerical method is taken to study the impacts of friction coefficient on brake groan. The calculation result shows that with the increase of kinetic friction coefficient /or the decrease of difference value between static friction coefficient and kinetic friction coefficient can prevent or restrain self-excited vibration from happening.

scholarly journals Experimental and Mathematical Model for the Antimalarial Activity of Methanolic Root Extract of Azadirachta indica (Dongoyaro) in Mice Infected with Plasmodium berghei NK65

2020 ◽  
pp. 68-82
Author(s):  
Adeniyi Michael Olaniyi ◽  
Momoh Johnson Oshiobugie ◽  
Aderele Oluwaseun Raphael
Keyword(s):  
Mathematical Model ◽  
Azadirachta Indica ◽  
Plasmodium Berghei ◽  
Antimalarial Activity ◽  
Equilibrium Points ◽  
Standard Procedure ◽  
Root Extract ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Group B

The study determines the experimental and mathematical model for the anti-plasmodial activity of methanolic root extract of Azadirachta indica in Swiss mice infected with Plasmodium berghei NK65. Phytochemical analyses, antimalarial activity of the methanolic root extract of A. indica was determined in mice infected with Plasmodium berghei NK65 using standard procedure. Liver biomarker enzymes were also determined. The model P. berghei induced free and P. berghei infected equilibrium were determined. The stability of the model equilibrium points was rigorously analyzed. The phytochemicals present in the extract include: alkaloid, flavonoid, saponin and phenolic compounds etc. The experimental study consists of five groups of five mice each per group. Group A, B, C, D and E were healthy, infected without treatment, infected mice treated with fansidar (10 mg/kg), chloroquine (10 mg/kg) and 250 mg/kg body weight of A. indica methanolic root extract respectively. The extract showed anti-plasmodial activity of 73.96%. The result was significant when compared with group B mice, though it was lower than that exhibited by fansidar (88.91%) and chloroquine (92.18%) for suppressive test. There were significant decrease (P<0.05) in plasma AST and ALT levels in the treated infected mice compared to the infected untreated mice. The results of the model showed that the P.berghei induced free equilibrium is locally and globally asymptotically stable at threshold parameter,  less than unity and unstable when  is greater than unity. Numerical simulations were carried out to validate the analytic results which are in agreement with the experimental analysis of this work.

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